Find the coordinates of the point which divides the join of and in the ratio .

Coordinate Geometry | Exercise 7.2

Question 1

Find the coordinates of the point which divides the join of $(-1, 7)$ and $(4, -3)$ in the ratio $2 : 3$ .

Solution

Section Formula: If a point $\text{P}(x, y)$ divides the line segment joining $\text{A}(x_1, y_1)$ and $\text{B}(x_2, y_2)$ in the ratio $m:n$ internally, then:

$x = \frac{mx_2 + nx_1}{m + n}, \quad y = \frac{my_2 + ny_1}{m + n}$

The ratio $m:n$ tells us how the point splits the segment — $m$ parts from A to P, and $n$ parts from P to B. Here the ratio is $2:3$ , so P is closer to A.

We use the section formula to find the coordinates of the dividing point.

Step 1 — Identify the values

Let's identify the given coordinates. Let the first point be $P_1 = (x_1, y_1) = (\mathbf{-1}, \mathbf{7})$ . Let the second point be $P_2 = (x_2, y_2) = (\mathbf{4}, \mathbf{-3})$ . The given ratio is $m_1 : m_2 = \mathbf{2} : \mathbf{3}$ .

Step 2 — Calculate the x-coordinate

We apply the section formula for the x-coordinate. $x = \frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}$

$x = \frac{\mathbf{2} \times \mathbf{4} + \mathbf{3} \times (\mathbf{-1})}{\mathbf{2} + \mathbf{3}}$

$x = \frac{8 - 3}{5}$

$x = \frac{5}{5}$

$\boxed{x = 1}$

Step 3 — Calculate the y-coordinate

Now, we apply the section formula for the y-coordinate. $y = \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2}$

$y = \frac{\mathbf{2} \times (\mathbf{-3}) + \mathbf{3} \times \mathbf{7}}{\mathbf{2} + \mathbf{3}}$

$y = \frac{-6 + 21}{5}$

$y = \frac{15}{5}$

$\boxed{y = 3}$

Answer

(i) The coordinates of the point are $\mathbf{(1, 3)}$ .

More questions in Exercise 7.2

Find the coordinates of the point which divides the join of $(-1, 7)$ and $(4, -3)$ in the ratio $2 : 3$ .

Find the coordinates of the points of trisection of the line segment joining $(4, -1)$ and $(-2, -3)$ .

To conduct Sports Day activities, in your rectangular shaped school ground ABCD, lines have been drawn with chalk powder at a distance of 1m each. 100 flower pots have been placed at a distance of 1m from each other along AD, as shown in Fig. 7.12. Niharika runs $\frac{1}{4}$ th the distance AD on the 2nd line and posts a green flag. Preet runs $\frac{1}{5}$ th the distance AD on the eighth line and posts a red flag. What is the distance between both the flags? If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags, where should she post her flag?

Find the ratio in which the line segment joining the points $(-3, 10)$ and $(6, -8)$ is divided by $(-1, 6)$ .

Find the ratio in which the line segment joining $A(1, -5)$ and $B(-4, 5)$ is divided by the $x$ -axis. Also find the coordinates of the point of division.

If $(1, 2)$ , $(4, y)$ , $(x, 6)$ and $(3, 5)$ are the vertices of a parallelogram taken in order, find $x$ and $y$ .

Find the coordinates of a point A, where AB is the diameter of a circle whose centre is $(2, -3)$ and B is $(1, 4)$ .

If A and B are $(-2, -2)$ and $(2, -4)$ , respectively, find the coordinates of P such that $\text{AP} = \frac{3}{7} \text{AB}$ and P lies on the line segment AB.

Find the coordinates of the points which divide the line segment joining $A(-2, 2)$ and $B(2, 8)$ into four equal parts.

Q10

Find the area of a rhombus if its vertices are $(3, 0)$ , $(4, 5)$ , $(-1, 4)$ and $(-2, -1)$ taken in order.

[Hint : Area of a rhombus = $\frac{1}{2}$ (product of its diagonals)]